Control of Discrete-Time PartiallyObserved Jump Linear Systems
Over Causal Communication
Systems
C. D. Charalambous
Depart. of ECE
University of Cyprus
Nicosia, Cyprus
CDC 2006, San Diego
S. Z. Denic
Depart. of ECE
University of Arizona
Tucson
1
Control Over Communication Channel


Block diagram of a control-communication problem
The source is partially observed jump system and
communication channel is causal
Dynamical
System
Sensor
Encoder
Collection and Transmission
of Information
(Node 1)
CDC 2006, San Diego
Communication
Channel
Capacity Limited
Link
Decoder
Sink
Reconstruction with
Distortion Error (Node 2)
2
Critical Features

Critical features from the communication and
control point of view

Amount of data produced by a particular source (sensor)
– source entropy
Capacity of the communication channel
Controllability and observability of the controlled system


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References [Plenty More]
Nair, Dey, and Evans,“Communication limited
stabilisability of jump Markov linear systems,” In Proc.
15th Ini. Symp.Math. The. New. Sys., U. Notre Dame,
USA, Aug 2002.
 Nair, Dey, and Evans, “Infimum data rates for
stabilising Markov jump linear systems,” in Proc. 42th
IEEE Conf Dec. Contr., pp. 1176-1181, 2003.
 C. D. Charalambous, “Information theory for control
systems: causality and feedback,” in Workshop on
Communication Networks and Complexity, Athens,
Greece, August 30-September 1, 2006.

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Overview


Problem formulation
Causal communication channels and systems







Mutual information for causal channels
Data processing inequalities for causal communication channels
Capacity for causal communication channels
Rate distortion for causal communication channels
Information transmission theorem
Necessary conditions for observability and stabilizability
over causal communication channels
Conclusions
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Problem Formulation
Problem
formulation
Causal communication channels and systems
Necessary conditions for observability and
stabilizability over causal communication channels
Conclusions
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Problem Formulation

Block diagram of control/communication system
X t 1  A  St  X t  B  St Wt  N  St U t , X 0  X
Yt  C  St  X t  D  St Vt
St t 0 , St   1,..., M 


Pr St 1   j | St   i  pij
Wt  N  0, I k 
Vt  N  0, I l 
X 0  N  x0 , Q0 
Wt ,Vt , St , X 0 : t  N 0 
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independent
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Problem Formulation

Encoder, Decoder, Controller are causal


Z t  c Y0t , Z 0t 1, Z 0t 1, S0t 
Yt  d  Z 0t , Y0t 1, S0t 
U t   Y0t ,U 0t 1, S0t

with feedback
Communication channel causality

 

P dZ t | z0n , z0t 1  P dZ t | z0t , z0t 1 , n  t
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Problem Formulation

System performance measures
Definition 2.1: (Observabilit in Probability). The system is observable in probability if
for any D, δ ≥ 0 there exist an encoder and decoder such that
1 t 1
lim  Pr Yk  Yk    D,
t  t
k 0


Definition 2.2: (Observability in r-th mean). The system is observable in r-th mean if
there exist an encoder and decoder such that
r
1 t 1 
lim  E Yk  Yk   D, r  0
t  t


k 0
where D ≥ 0 is finite.
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Problem Formulation

System performance measures
Definition 2.3: (Stabilizability in probability). The system is stabilizable in
probability if for any D, δ ≥ 0 there exist a controller, encoder and decoder
such that
1 t 1
lim  Pr  X k  0     D,
t  t
k 0
Definition 2.4: (Stabilizability in r-th mean). The system is asymptotically
stabilizable in r-th mean if there exist a controller, encoder and decoder
such that
1 t 1 
r
lim  E X k  0   D, r  0


t  t
k 0
where D ≥ 0 is finite.
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Causal Communication
Channels and Systems
Problem
formulation
Causal communication channels and systems
 Mutual information for causal channels
 Data processing inequalities for causal communication
channels
 Capacity for causal communication channels
 Rate distortion for causal communication channels
 Information transmission theorem
Necessary conditions for observability and stabilizability over
causal communication channels
Conclusions
CDC 2006, San Diego
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Causal Communication Channels and
Systems

Lemma 3.2: Let



iR Z 0T 1; Z 0T 1  log

p z0T 1 | z0T 1

p z0T 1


denote the self-mutual information when the RND

p z0T 1 | z0T 1

p z0T 1


is restricted to a non-anticipative or causal feedback channel with memory. Then, the
restricted mutual information is given by





T
T 1 T 1
T

1
T

1
   I Z i ; Z | Z i 1
IC Z0 ; Z0
 E i R Z 0 ; Z 0
0 i
0

 i 0

Directed Information
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Causal Communication Channels and
Systems

Remark: In general, causal mutual information is not
symmetric



IC Z0T 1; Z0T 1  I C Z 0T 1; Z 0T 1


Data processing inequality for causal channels



 
 
I Z 0n ; Z 0n  I C Z 0n ; Z 0n  I Y0t ; Z 0n  I Y0t ;Y0t



 I C Y0t ;Y0t , n, t  N 0
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Causal Communication Channels and
Systems

Channel capacity based on the causal mutual
information

1
1
CC  lim CT  lim
sup IC Z0T ; Z 0T
T  T
T  T p T 
Z


0
Rate distortion based on the causal mutual information

1
1
RT  D   lim
inf
I C Y0T ;Y0T
T  T
T  T p T T M
Y0 |Y0
RC  D   lim
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
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Causal Communication Channels and
Systems

Theorem 4.1: (Information Transmission Theorem)
Suppose the different communication blocks in Fig. 1 form
a Markov chain. Consider a control-communication
system where the communication channel is restricted to
being causal. A necessary condition for reconstructing a
source signal Yt up to a distortion level D from Z t is given
by
RC  D   CC
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Necessary conditions for
observability and stabilizability over
causal communication channels
Problem
formulation
Causal communication channels and systems
 Mutual information for causal channels
 Data processing inequalities for causal communication
channels
 Capacity for causal communication channels
 Rate distortion for causal communication channels
 Information transmission theorem
Necessary conditions for observability and stabilizability over
causal communication channels
Conclusions
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Necessary conditions for observability and
stabilizability over causal communication channels

Lemma 4.2. Consider the following single letter distortion measure
1 T
 (Yi  Yi ) : R p  [0, )
T (Y0T ,Y0T )    (Yi  Yi ) , where
T i 0
Then, a lower bound for
1
RT ( D )
T
is given by
1
1
RT ( D)  H S (Y0T )  max H S (h),
T
T
hGD
p
where GD  {h : R  [0, );  h( y )dy  1,   ( y )h( y )dy  D}.
Rp
Rp
It follows
RC ( D )   (  0 )  H S (h* )
and under some conditions, this lower bound is exact for D  0.
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Necessary conditions for observability and
stabilizability over causal communication channels
Theorem
4.3. Consider a jump control-communication
system where Yt  R p is the observed process at time t. Let pS   s 
be a steady state distribution of the underlying Markov chain.
Introduce the following notation


X t  E  X t |  Y0t 1,U0t 1, S0t 1 


Q  lim Qt
Xt
Qt


E  X t X t tr |  Y0t 1,U 0t 1, S0t 1 


t 
A necessary condition for asymptotic observability and
stabilizability in probability is given by
CC 
Mp
1
log  2 e   log[(2 e) p det  g ]
2
2


  log det C  i  Q,i C tr  i   D  i  D tr  i  p   i 
S
i
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Necessary conditions for observability and
stabilizability over causal communication
channels
g is the covariance matrix of the Gaussian
distribution h* ( y ) ~ N (0,  g ),( y  R p ) which satisfies

|| y||
h* ( y )dy  D.
A necessary condition for asymptotic observability
and stabilizability in r-th mean is given by
p
Mp
CC 
log  2 e   log e r  log(
2

p
p
( ) r ).
r
p
dVd ( ) rD
r

  log det C  i  Q,i C tr  i   D  i  D tr  i  p   i 
S
i
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Conclusion

General necessary conditions for observability and stabilizability for
jump linear systems controlled over a causal communication
channel are derived.

Causal Information Theory is Essential for Channels with Feedback
and Memory

Different criteria for observability and stabilizability corresponds to
different necessary condition.

Future work

Sufficient conditions (design encoders and decoders)
 Channel-source matching
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